EXAMPLE 4 Solve equations using nth roots

Solve the equation.

a. 4x5 = 128

Divide each side by 4.x5 32=

Take fifth root of each side.x = 325

Simplify.x 2=

EXAMPLE 4 Solve equations using nth roots

b. (x – 3)4 = 21

Take fourth roots of each side.

x – 3 4+– 21=

Add 3 to each side.x 4+– 21 + 3=

Write solutions separately.

x 421 + 3= or x = 421 + 3–

Use a calculator.x 5.14 or x 0.86

EXAMPLE 5 Use nth roots in problem solving

Biology

A study determined that the weight w (in grams) of coral cod near Palawan Island, Philippines, can be approximated using the model

w = 0.0167ℓ 3

where l is the coral cod’s length (in centimeters). Estimate the length of a coral cod that weighs 200 grams.

EXAMPLE 5 Use nth roots in problem solving

SOLUTION

Write model for weight.w 0.0167 l 3=

Substitute 200 for w.200 0.0167 l 3=

Divide each side by 0.0167.11,976 l 3

Take cube root of each side.11,9763 l

Use a calculator.22.9 l

ANSWER A coral cod that weighs 200 grams is about 23 centimeters long.

GUIDED PRACTICE for Examples 4 and 5

Solve the equation. Round the result to two decimal places when appropriate.

13. x3 = 64

x3 = 64

take 3rd root of each side. x = 3 64

Simplify. x = 4

SOLUTION

GUIDED PRACTICE for Examples 4 and 5

14.

12

x5 = 512

SOLUTION

12

x5 = 512

Multiply each side by 2.x5 = 1024

take 5th root of each side. x = 5 1024

Simplify.x = 4

GUIDED PRACTICE for Examples 4 and 5

15. 3x2 = 108

SOLUTION

3x2 = 108

Divide each side by 3.x2 = 36

Simplify.x –= + 6

take 2nd root of each side. x = 2 36

GUIDED PRACTICE for Examples 4 and 5

16. 14

x3 = 2

SOLUTION

14

x3 = 2

Multiply each side by 4.x3 = 8

take 3rd root of each side. x = 3 8

Simplify.x = 2

GUIDED PRACTICE for Examples 4 and 5

17. ( x – 2 )3 = –14

SOLUTION

( x – 2 )3 = –14

take 3rd root of each side. ( x – 2 ) = 3 –14

add 2 to both sides. x = 3 –14 + 2

Write solution.x = 3 –14 + 2

x = – 0.41 Use a calculator.

GUIDED PRACTICE for Examples 4 and 5

18. ( x + 5 )4 = 16

SOLUTION

( x + 5 )4 = 16

take 4th root of each side. ( x + 5 ) = + 4 16

add 5 to each side. x = + 4 16 – 5

Write solutions separately. x = 2 – 5 or x = – 2 – 5

Use a calculator.x = – 3 or x = –7

GUIDED PRACTICE for Examples 4 and 5

WHAT IF? Use the information from Example 5 to estimate the length of a coral cod that has the given weight.

19.

a. 275 grams

SOLUTIONWrite model for weight.w = 0.0167l 3

Substitute 275 for w.275 = 0.0167l 3

Divide each side by 0.0167.16467 = l 3

Take cube root of each side. 3 16467 = l

Use a calculator.25 = l

GUIDED PRACTICE for Examples 4 and 5

A coral cod that has the 275 grams is about 25 cm long.

ANSWER

GUIDED PRACTICE for Examples 4 and 5

b. 340 grams

SOLUTIONWrite model for weight.w = 0.0167l 3

Substitute 340 for w.340 = 0.0167l 3

Divide each side by 0.016720360 = l 3

Take cube root of each side 3 20360 = l

Use a calculator27 = l

A coral cod that has the 340 grams is about 27 cm long.

ANSWER

GUIDED PRACTICE for Examples 4 and 5

c. 450 grams

SOLUTIONWrite model for weight.w = 0.0167l 3

Substitute 450 for w.450 = 0.0167l 3

Divide each side by 0.016726946 = l 3

Take cube root of each side 3 26946 = l

Use a calculator30 = l

A coral cod that has the 450 grams is about 30 cm long.

ANSWER